A review of compactness methods for cross-diffusion systems seen as Wasserstein gradient flows

TL;DR

This paper reviews methods for proving the existence of gradient flows in cross-diffusion systems using Wasserstein metrics, highlighting techniques like the Jordan-Kinderlehrer-Otto scheme.

Contribution

It provides a comprehensive overview of compactness methods and their application to various cross-diffusion systems, including new insights and literature synthesis.

Findings

Established existence of gradient flows for cross-diffusion systems.

Applied flow interchange and gradient inequality techniques.

Illustrated methods on parabolic and hyperbolic-parabolic systems.

Abstract

A comprehensive methodology for establishing the existence of gradient flows for cross-diffusion systems with respect to suitable energies is proposed. The approach is based on the construction of piecewise-in-time constant approximations via the Jordan-Kinderlehrer-Otto scheme. Compactness of the approximate sequence is obtained using either the flow interchange technique or the five gradient inequality. These methods are illustrated for both parabolic and hyperbolic-parabolic Busenberg-Travis systems, as well as for several of their variants. This paper reviews the results from the literature and discusses additional properties.